Coupled fermion–kink system in Jackiw–Rebbi model

In this paper, we study Jackiw–Rebbi model, in which a massless fermion is coupled to the kink of \(\lambda \phi ^4\) theory through a Yukawa interaction. In the original Jackiw–Rebbi model, the soliton is prescribed. However, we are interested in the back-reaction of the fermion on the soliton besides the effect of the soliton on the fermion. Also, as a particular example, we consider a minimal supersymmetric kink model in (\(1+1\)) dimensions. In this case, the bosonic self-coupling, \(\lambda \), and the Yukawa coupling between fermion and soliton, g, have a specific relation, \(g=\sqrt{\lambda /2}\). As the set of coupled equations of motion of the system is not analytically solvable, we use a numerical method to solve it self-consistently. We obtain the bound energy spectrum, bound states of the system and the corresponding shape of the soliton using a relaxation method, except for the zero mode fermionic state and threshold energies which are analytically solvable. With the aid of these results, we are able to show how the soliton is affected in general and supersymmetric cases. The results we obtain are consistent with the ones in the literature, considering the soliton as background.

Coupled fermion–kink system in Jackiw–Rebbi model

Eur. Phys. J. C
Coupled fermion-kink system in Jackiw-Rebbi model
A. Amado 0
A. Mohammadi 0
0 Departamento de Física, Universidade Federal de Pernambuco , Recife, PE 52171-900 , Brazil
In this paper, we study Jackiw-Rebbi model, in which a massless fermion is coupled to the kink of λφ4 theory through a Yukawa interaction. In the original Jackiw-Rebbi model, the soliton is prescribed. However, we are interested in the back-reaction of the fermion on the soliton besides the effect of the soliton on the fermion. Also, as a particular example, we consider a minimal supersymmetric kink model in (1 + 1) dimensions. In this case, the bosonic self-coupling, λ, and the Yukawa coupling between fermion and soliton, g, have a specific relation, g = √λ/2. As the set of coupled equations of motion of the system is not analytically solvable, we use a numerical method to solve it self-consistently. We obtain the bound energy spectrum, bound states of the system and the corresponding shape of the soliton using a relaxation method, except for the zero mode fermionic state and threshold energies which are analytically solvable. With the aid of these results, we are able to show how the soliton is affected in general and supersymmetric cases. The results we obtain are consistent with the ones in the literature, considering the soliton as background.
1 Introduction
Solitons, named by Zabusky and Kruskal [
1
], first appeared
as a solution for KdV equation [
2
]. They play important roles
in diverse areas of physics, biology and engineering [
3–5
].
In one spatial dimension kinks, and in higher spatial
dimensions vortices, monopoles, instantons and domain walls, are
amongst the most important ones in this category. They are
topological configurations which appear in different areas of
physics such as high energy, atomic and condensed matter
physics [
6–10
]. These topologically nontrivial configurations
cannot be continuously deformed into a trivial vacuum
configuration.
The coupling of the fermionic field to other fields alters
the energy spectrum of the fermionic field as well as the wave
function which can cause many interesting phenomena.
Models including coupled fermionic and bosonic fields are
crucial in many branches of physics, specially when the bosonic
field has the form of a soliton. As solitons can be viewed as
extended particles with finite mass, i.e. finite energy at rest,
the systems consisting of coupled fermionic and solitonic
fields can be good candidates to describe extended objects
such as hadrons. Since 1958, with Skyrme’s pioneering work
[
11–15
], many physicists have tried to explain the hadrons
and their strong interactions nonperturbatively using
phenomenological nonlinear field theories [
16–18
].
The presence of a soliton in fermion–soliton systems
distorts the fermion vacuum state and consequently can
induce nonzero vacuum polarization and Casimir energy. The
corresponding nontrivial topology induces nonzero vacuum
expectation values of physical observables. These
phenomena have been widely discussed in the literature for different
types of solitons and in different dimensions (e.g. [
19–24
]).
Moreover, when a fermion interacts with a soliton, an
interesting phenomenon occurs which is the assignment of
fractional fermion number to the solitonic state. In a theory where
all the fields carry integer quantum numbers, the emergence
of fractional quantum numbers has attracted much interest.
Jackiw and Rebbi pointed out the occurrence of the fractional
fermion number for the first time [25]. Much of the work in
this area has been inspired by the Jackiw and Rebbi’s
pioneering work. They considered some models with the charge
conjugation symmetry where the fermion is coupled to a bosonic
background field in the form of a soliton. They have shown
that the existence of a fermionic non-degenerate zero mode
implies the soliton with fermion number one half [
3,26
].
Coupled fermion–soliton systems also appear in the
braneworld scenarios in the context of the localization of the
Standard Model fields on the brane. Localising a fermion on
the brane was first described by the original work [
27
], where
our Universe can be realized inside a domain wall embedded
in a (4+1)-dimensional world. The localization of spin 1/2
fermions on thin branes due to a soliton, is performed via
the mechanism introduced by Jackiw and Rebbi [
25
]
originally to demonstrate the fermion charge-fractionization
phenomenon. Also, higher-dimension extensions for this
mechanism have been studied in the literature [
28–31
]. Besides
that, the localization of fermions on a double-brane in warped
space-time has been studied (e.g. [32]).
In most of the models investigated in the literature
consisting of coupled fermion–soliton systems, the soliton is
considered as a background field. The main reason is that solving
the nonlinear system treating both fields as dynamical1 is in
general extremely difficult analytically [
33
]. There have been
some attempts to do so that were successful only in specific
points of the parameter space, in practice close to the
background soliton case [
34,35
]. In principle, the soliton in these
systems can have infinitely different shapes. Based on Jackiw
and Rebbi’s work [25], the back-reaction of the fermion on
the soliton is small when the coupling of the interaction term
is small. Therefore, in this regime considering the soliton
to be a prescribed field is a good approximation, although
when the coupling is not small it may fail considerably. In
Jackiw–Rebbi model, the system has charge conjugation and
energy-reflection symmetries. The energy-reflection
symmetry relates each fermionic mode with energy E to the one
with energy −E , which makes the energy spectrum to be
completely symmetric with respect to the line E = 0.
In this paper, we study the Jackiw–Rebbi model with a
massless fermion coupled to the kink of λφ4 theory as well
as a minimal supersymmetric kink model, as an example,
and solve the system using a numerical method. As we solve
the system self-consistently, it is possible to analyse not only
the effect of the soliton on the fermion field but also the
back-reaction of the fermion field on the soliton. In this case,
the non-degenerate zero mode is always present regardless
of the values of the parameters of the model (the same
happens in Jackiw–Rebbi model), and the soliton receives no
back-reaction due to this mode. In the dynamical model, the
system loses the charge conjugation and energy-reflection
symmetries, resulting in a nonsymmetrical energy spectrum
with respect to the E = 0 line. We show that our results
are consistent with the ones discussed in the literature with
the prescribed soliton, where the system recovers its charge
conjugation and energy-reflection symmetries.
This paper is organized into five sections: in Sect. 2 we
briefly introduce the fermion–kink model in (1 + 1)
dimensions as well as the formulation of the problem. In this
sec1 In this paper we use the word dynamical, in contrast to prescribed, to
refer to the result of the equations of motion considering both fermionic
and bosonic fields together. In the prescribed model we consider the
soliton to be a prescribed (or background) field that does not receive
any back-reaction from the fermion.
tion, we write the Lagrangian describing our model, in
components, and the resultant equations of motion. In Sect. 3
we obtain the fermionic bound states and bound energies of
the system. We find the fermionic zero mode and threshold
bound energies analytically. To obtain the other fermionic
bound energies, bound states and the shape of the soliton we
use a numerical method called relaxation method. At the end
of the section, we show the classical soliton mass as well as
the back-reaction of the fermion on the soliton. In Sect. 4 we
solve the system for the particular case of the supersymmetric
kink, as an example. Finally, Sect. 5 is devoted to a summary
and a discussion of our results.
2 Fermion–kink system
We consider a fermion–soliton system in (1+1) dimensions
given by the following Lagrangian:
1 1
L = 2 ∂μφ∂μφ + 2 ψ¯ i γ μ∂μψ − g φ ψ¯ ψ − V (φ),
(2.1)
with V (φ) = λ4 φ2 − mλ2 2, the known φ4 theory potential.
In order to guarantee a well-defined energy for the soliton, we
have φ (x → ±∞) = ± √mλ ≡ ±φ0. Considering the field φ
to be static, the Euler–Lagrange equations of the system are
given by
− i γμ∂μψ + 2g φ ψ = 0,
φ − λφ φ2 −
λ
m2
− gψ¯ ψ = 0,
ψ = e−i Et
ψ1 , these equations become
ψ2
where prime denotes differentiation with respect to x . In
this work, we consider the fermion field as a classical
cnumber Dirac wave function. Defining χ ≡ φ/φ0 and
E ψ1 + ψ2 − 2g φ0 χ ψ2 = 0,
E ψ2 − ψ1 − 2g φ0 χ ψ1 = 0,
− χ
+ m2χ (χ 2 − 1) + 2g/φ0 ψ1ψ2 = 0.
in which the representation for the Dirac matrices is chosen as
γ 0 = σ1, γ 1 = i σ3 and γ 5 = σ2. Without loss of generality,
we choose the components ψ1 and ψ2 as real since there
are only two independent degrees of freedom in ψ . Also,
we rescale all the quantities to dimensionless ones as ψ →
√mψ , χ → χ , E → m E , φ0 → φ0 (λ → m2λ), g → mg
and x → x /m.2
2 We write the mass dimension in parentheses when deemed necessary
to avoid confusion.
(2.2)
(2.3)
As can be seen in the Lagrangian (2.1), the fermion
field interacts nonlinearly with the pseudoscalar field. The
system cannot be solved analytically without imposing the
soliton to be a background field. Thus, using a numerical
method, we solve this coupled set of differential equations
self-consistently and find the fermionic bound states and
bound energies as well as the shape of the soliton. The shape
of the static soliton in the model we consider here is not
prescribed and is determined by the equations of motion.
With this, besides the effect of the soliton on the fermion, we
can obtain at the classical level the effect of the fermion on
the soliton (the back-reaction), within our numerical
restrictions. The main advantage is to help us understand the
system beyond the regime considered in the literature where the
soliton can be treated as background which is equivalent to
g/φ0 → 0 limit.
In the limit g/φ0 → 0 (g → 0 and/or λ → 0), the last
equation in (2.3) decouples from the others and has analytical
solution, i.e. the kink of λ φ4 theory. The solutions of some
similar systems in this limit have been studied in detail in
[
25,36
]. In this limit, the solutions for this equation are
χbg(x ) = ± tanh
x − x0
√2
.
In this paper, we consider the positive sign in Eq. (2.4). One
can calculate the classical soliton mass using the expression
Mcl =
∞
−∞
that is, Mcl = 2 √32 m φ02 for the kink of λ φ4 theory. We
solve the set of coupled equations (2.3) self-consistently and
check the results with the ones considering the soliton as
background.
As in g/φ0 → 0 limit the solitonic equation decouples
from the fermionic one, the system retrieves charge
conjugation (C ) and energy-reflection (R) symmetries. To see this
point we can rewrite the fermionic equation of motion in the
following form:
(−i γ 0γ i ∂i − 2g φ γ 0)ψE = E ψE,
where we separated the time and space components. If the
system has R symmetry this should be equivalent to
(−i γ 0γ i ∂i − 2g φ γ 0) RψE = −E RψE,
leading to Rγ 0 = −γ 0 R and Rγ 0γ i = −γ 0γ i R. These
conditions are respected if we take R = γ 1. The existence of
this symmetry implies that the negative and positive energy
spectra are mirror images of each other around E = 0, in
this limit. We can proceed in a similar manner to show that
(2.4)
(2.5)
(2.6)
(2.7)
(3.1)
(3.2)
the system possesses charge conjugation symmetry, which
in the representation adopted here is C = σ3, along with
considering the complex conjugation of the wave function.
However, as the ratio g/φ0 increases, the gψ¯ ψ term in
second equation of (2.2) cannot be neglected anymore and
breaks these symmetries. As a result, one can see that the
energy spectrum is not symmetric around E = 0 in
general. In [
36
] the authors have obtained a symmetric
spectrum around E = 0. This is because in the system they
have considered, the soliton is a background field no
matter how big the ratio g/φ0 could be. Therefore, the result
obtained in the referred paper is not a good approximation
for the model described by the Lagrangian (2.1) when the
ratio g/φ0 is not small enough to consider the prescribed
soliton. In this paper, we compare our results with the ones
with background soliton. Besides these two discrete
symmetries, the system displays parity symmetry ( P), represented
by the operator P = γ 1, together with the transformation
x → −x . Although the term gψ¯ ψ in the equation of motion
breaks the charge conjugation and energy-reflection
symmetries, it does not break parity symmetry. This way, the
system has parity and as a result the wave functions display this
symmetry regardless of the value of g/φ0. This feature can
be seen shortly in our numerical results.
3 Bound states and bound energies
We obtain the zero energy bound state and threshold energies
analytically, although to find the other bound states we have
to rely on a numerical method.
3.1 Zero energy bound state
For the zero energy bound state the equations of motion are
simplified and we are able to obtain the analytical solution
of the system in the whole g and φ0 intervals. Taking E = 0,
the equation system becomes
ψ1 + 2gφ0 χ ψ1 = 0,
ψ2 − 2gφ0 χ ψ2 = 0,
− χ
+ χ (χ 2 − 1) + 2g/φ0 ψ1ψ2 = 0.
It turns out that the first two equations can easily be solved
as functions of χ , yielding
ψ1(x ) = a1 e−2gφ0 1x χ(x ) dx ,
ψ2(x ) = a2 e2gφ0 1x χ(x ) dx .
Notice that if we define f (x ) ≡ e−2gφ0 1x χ(x ) dx , either
f (x → ±∞) = 0 and consequently f −1(x → ±∞)
diverges or f −1(x → ±∞) = 0 and as a result f (x → ±∞)
diverges. As the normalization of the divergent components
should be zero, we can conclude either a1 = 0 or a2 = 0.
Thus, the term with ψ -dependence in the last equation
vanishes and we find
φ0
It is important to notice that based on this result, the
backreaction of the fermion on the soliton is zero for the fermionic
zero mode. This is consequence of the energy-reflection
symmetry in the first equation of (2.2). It can be shown using the
following relation:
ψ¯ EψE = ψE†γ 0ψE = ψE† R† Rγ 0ψE = −(RψE)†γ 0(RψE)
= −(Rψ E)(RψE)
(3.5)
knowing the facts R† R = 1 and Rγ 0 = −γ 0 R as we
mentioned before. For the zero energy, one can use Rψ0 = αψ0
and |α|2 = 1, which gives ψ¯ EψE = 0 [
33,37
]. This means
that the back-reaction is zero for the fermionic zero mode. If
a system does not possess this symmetry, the back-reaction
for the fermionic zero mode is nonzero in general [33].
3.2 Threshold states
Threshold or half-bound states are the states where the
fermion field goes to a constant at spatial infinity. For these
states when x → ∞ the wave function is finite but does
not decay fast enough to be square-integrable [
38–40
]. We
are interested in the energies associated with these states in
order to show the division between the bound and continuum
energy spectrum.
To find such states in our system, we solve the system of
equations at x → ±∞. We write ψ in the form ψ (x →
±∞) = e−i Et c1 , where the ci are arbitrary constants.
c2
Applying the conditions χ (x → ±∞) = ±1, χ (x →
0
±∞) = χ (x → ±∞) = 0 and ψ (x → ±∞) = 0 to
the set of equations of motion, we obtain
(3.6)
This set of equations has a nontrivial solution only when the
last equation decouples, i.e. g/φ0 → 0. Solving the first two
equations in this case, it is easy to show that the energies of
the threshold states are E = ±2gφ0, as expected.
3.3 Numerical method
The remaining bound states cannot be found analytically and
a numerical method is required. We use a relaxation method
that starts with an initial guess and iteratively converges to
the solution of the system. We start with the known energy
spectrum and bound states where the soliton is a background
field [
36
] and find the solution of the system, considering
φ0 = π , which gives a soliton with winding number one.
There are two first-order and one second-order differential
equations (Eq. (2.3)). The latter can be transformed to a set
of two first-order equations as
p ≡ χ ,
2g
− p + χ (χ 2 − 1) + φ0 ψ1ψ2 = 0.
(3.7)
To find the fermion energy eigenvalue, we also use the
equation E = 0, reflecting the fact that energy is constant.
Moreover, we fix the translational symmetry of the system by
choosing x0 = 0.
Now, there are five coupled first-order differential
equations which need five boundary conditions. Among the
several boundary conditions available, we choose the following:
χ (±∞) = ±1, χ (0) = 0, ψ1(±∞) = 0.
(3.8)
In Fig. 1 we show the energy spectrum as a function of
the coupling g for the soliton with winding number one,
φ0 = π . The left graph shows the first three positive and
negative energy levels of the system. In the middle and
right graphs, the positive and negative energy levels,
respectively, are zoomed in specific regions of the parameters. We
have depicted our result with the solid curves and compared
with the dashed ones, the background soliton results. As one
can see, our results and the ones with the background kink
become more different as g grows. Also, the symmetry of
the energy levels around the line E = 0, expected in the
background model, breaks gradually when we increase g
from zero. This becomes evident noticing that both positive
and negative energies are lower than their counterparts in the
background model.
As a measure of the back-reaction of the fermion on
the soliton, we calculate the root mean squared deviation
1.5
2.
gies in the dynamical and background model, respectively. Right graph:
solid and dashed curves depict negative fermionic bound energies in
the dynamical and background model, respectively. In all graphs dotted
lines depict the fermionic threshold energies in g/φ0 → 0 limit
0.2
for the first two positive energy levels and the dashed ones for the first
two negative energy levels. The dotted line corresponds to the soliton
mass in the background model. The two black dots are the points with
specific values of g for which the represented bound states first appear
4 An example: supersymmetric kink model
We consider the minimal supersymmetry in a (1+1)
dimensional field theory. The supersymmetric Lagrangian has the
form [
41, 42
]
in which φ is considered static. Note that the supersymmetry
relates the bosonic self-coupling λ and the Yukawa
interaction coupling g.
Taking g = √λ/2 in the set of equations of motion (2.3),
we obtain
between the soliton in the dynamical model and the
background one, δRMS. In the left graph of Fig. 2, one can observe
that the back-reaction increases with the coupling g. In small
values of this coupling, the difference in the back-reaction
for positive and negative energy levels is low and as g grows
this difference increases. It reflects the fact that in g/φ0 → 0
limit the energy-reflection symmetry is present, being
gradually broken with increasing g, which distorts the symmetry
between the positive and negative energy levels.
As a final result, we show the soliton mass as a function of
the coupling g in the right graph of Fig. 2. We can see that by
increasing the coupling g, the soliton mass starts diverging
from the classical result, as expected.
L = 21 (∂μφ∂ μφ + ψ¯ i γ μ∂μψ + F2)
1
+ Wφ F − 2 Wφφ ψ¯ ψ,
where the subscript φ shows the derivative with respect to φ
and F is an auxiliary field. Using the Euler–Lagrange
equations and choosing the bosonic potential to be the kink
potential V (φ) = λ4 φ2 − mλ2 2, i.e. V (φ) = 21 Wφ2, we obtain
1 1
Lkink = 2 ∂x φ∂ x φ + 2 ψ¯ i γ μ∂μψ
−
λ λ
2 φ ψ¯ ψ − 4
φ2 − λ
depict the fermionic threshold energies in the background model. Right
graph: the derivative of the fermionic negative bound energy with
respect to φ0 as a function of φ0
In g/φ0 → 0 limit which is equivalent to λ → 0 (φ0 → ∞)
in the supersymmetric case, there are three fermionic bound
states with energies 0 and ±√3/2 (m) and two threshold ones
with energies E = ±√2 (m) [
36,42
].
We solve the system dynamically and discuss the case
where the soliton can be considered as background as well.
Again the zero energy bound state and threshold energies
can be obtained analytically, although to find the other bound
states we have to solve the system numerically.
For the zero mode, the equations of motion are simplified
and we are able to obtain the analytical solution of the system
in the whole φ0 interval. Using the same method as before
we observe that, interestingly, the solutions show to be φ0
independent. Requiring the wave function to be normalized,
we obtain
One can use the same method to find the threshold states
giving E = ±√2 (m).
To find the other bound states and the corresponding
parameters of the system we start with the known energy
spectrum and bound states in λ → 0 limit and solve the set
of equations for the whole region of φ0 within the numerical
restrictions. The same boundary conditions as in (3.8) are
considered.
The left graph in Fig. 3 shows the fermionic bound state
energies as a function of the asymptotic value of the bosonic
field, φ0. As can be seen, the background result is retrieved
as φ0 → ∞, i.e. E = ±√3/2 (m). It is important to
note that for φ0 2 the dynamical graphs and the lines
E = ±√3/2 (m) are not easily distinguishable, although for
0.2
0
0.5
smaller φ0 the negative and positive energies change
drastically from the λ → 0 (φ0 → ∞) limit result. In the numerical
simulations the closer φ0 is to zero, the more difficult it is
for the solutions to converge. The smallest values of φ0 we
are able to obtain are φ0 = 0.501 for the positive bound
energy and φ0 = 0.564 for the negative one, though based
on physical intuition it is possible to partially guess how the
energy curves would behave below these values. The
positive bound energy curve should not cross the zero energy
line as it would configure level crossing [
36
]. Furthermore,
as the negative energy curve becomes closer to the
threshold line E = −√2 (m), its slope decreases considerably at
φ0 ≈ 0.63, as the right graph of Fig. 3 shows.
Figures 4 and 5 depict the fermionic bound states as a
function of x for positive and negative energies, respectively.
The solid curves are the result of our dynamical model and
the dotted ones are the result in λ → 0 limit. We show the
results for two different low values of φ0 for positive and
negative energy states to highlight the effects in the region
far from λ → 0 limit. As can be seen in the graphs, in lower
φ0 case the dynamical and background results become more
distinct which confirms that in the low φ0 or large coupling
λ region the system cannot be described by the background
approximation.
To investigate the effect of the fermion on the shape of
the soliton, in Figs. 6 and 7 we show the bosonic field as a
function of x for positive and negative energies, respectively.
Since the results for positive energy change considerably in
low φ0 region, we show the result for three distinct values
of φ0 to make it possible to track the transition to the large
coupling λ limit. For each of the graphs, we show χ and its
spatial derivative, χ , to illustrate the way the soliton changes
from the background one. Interestingly enough, although the
slope of the kink of λφ4 theory is always positive, the
interaction with the fermion can be strong enough to invert the sign
of the slope at the origin. This seems to indicate that soliton is
assuming a kink–antikink–kink configuration as we increase
the coupling. In the large coupling region, the energy scale of
0 1
0 0.6
5
5
5
5
the fermion and the soliton mass become comparable. These
observations may mean that the fermion can create a kink–
antikink pair at large coupling. It is also important to notice
that although the soliton can change drastically in the large
λ region, the changes are limited to a small region around
the origin. This result shows that the back-reaction of the
fermion on the soliton and thus the disturbance region are
finite, except eventually for the limit φ0 → 0.
Using Eq. (2.5), the classical mass of the soliton is shown
in Fig. 8 for both positive and negative energy bound states in
the dynamical model as well as the background one. As can be
seen, for low φ0 the mass of the soliton diverges significantly
from the one in λ → 0 limit. However, for φ0 greater than 1
the three curves coincide within the scale shown in the graph.
Again, as a measure of the effect of the fermion on the
soliton, we calculate the root mean squared deviation between the
5
5
5
5
5
5
5
5
5
10
x
10
x
10
x
10
x
10
x
10
x
10
x
10
x
10
x
10
x
10
10
10
10
10
5
5
5
5
5
1
Soliton mass
15
10
5
0
0.5
1
0
0.5
1
1
2
3
Fig. 8 The classical soliton mass as a function of φ0. Solid and dashed
curves show the classical soliton mass in the dynamical model
corresponding to positive and negative bound energies, respectively. The
dotted line is the classical soliton mass for the background model
prescribed and dynamical soliton, δRMS. In Fig. 9, we show
this result as a function of φ0 and energy, for both positive and
negative bound states. As expected, the back-reaction of the
fermion on the soliton goes to zero when E → ±√3/2 (m),
i.e. λ → 0 results. Interestingly, the right graph in this figure
shows that the back-reaction decreases almost linearly with
energy. Also, the left graph of this figure confirms that when
φ0 goes to zero the back-reaction increases significantly and
cannot be neglected in this region, i.e. the large coupling
region.
5 Conclusion
In this paper, we have investigated a fermion–soliton model
in (1 + 1) dimensions in which a static pseudoscalar field
interacts nonlinearly with a Dirac particle. In this system, the
bosonic self-interaction part of the potential that is
responsible for creating a soliton with proper topological
characteristics has been considered to be the potential in λφ4 theory.
First, we have considered the general case where the Yukawa
coupling, g, is independent of the bosonic self-coupling, λ.
Then we have solved a minimal supersymmetric kink model
which is a particular example of the former with g = √λ/2.
We have found the zero mode fermionic state and
threshold energies analytically, although in order to find other
bound states and the corresponding shape of the soliton we
needed a numerical method. We used a relaxation method to
calculate the energy spectrum and the bound states as well
as the shape of the soliton. In the general case, where the
couplings g and λ are independent, we have solved the
system for the soliton with winding number one, i.e. φ0 = π .
As a consistency check, we have studied the limit where the
soliton can be considered as background, g/φ0 → 0 (λ → 0
in supersymmetric case).
Our calculations have shown that the back-reaction of the
fermion on the soliton for the fermionic zero mode is zero.
Therefore, the soliton corresponding to the fermionic zero
mode is the kink of λφ4 theory, even in the case where g/φ0
(λ in supersymmetric case) is large. This is due to the fact
that the first equation in 2.2 has energy-reflection symmetry,
which guarantees that the soliton receives no back-reaction
from the fermionic zero mode. Besides that, since the
energyreflection symmetry is broken for finite g/φ0, the energy
spectrum becomes progressively asymmetric around E = 0
with increasing g/φ0.
Our numerical results have shown that the energy
spectrum converges to the result of the background model as
g/φ0 → 0, unsurprisingly. The same happens with the
classical soliton mass. However, they are completely
distinguishable when g/φ0 is large. In the supersymmetric case by
varying the value of φ0 from zero to infinity, we could span the
region between g/φ0 → 0 and large g/φ0, within the
numerical limitations.
Furthermore, we have calculated the back-reaction of the
fermion on the soliton for the positive and negative energy
states as a function of φ0 and E for both general Jackiw–
Rebbi model and the supersymmetric case. The results show
RMS
2
1
0
1
2
3
4
0
2
RMS
2
1
0
2
E
Fig. 9 The back-reaction of the fermion on the soliton, δRMS, as a
function of φ0 and the fermionic bound energy E. Solid and dashed
curves show the back-reaction corresponding to the positive and
negative bound energies, respectively, in the dynamical model. Dotdashed
lines are the threshold energies in the background model
that the back-reaction of the fermion on the soliton tends
to zero as g/φ0 → 0 for both positive and negative bound
energy curves, as expected. In contrast, with large g/φ0 the
back-reaction increases significantly. When the value of g/φ0
is high enough it can distort the shape of the soliton to the
point that the slope of the soliton at the origin becomes
negative, even though for the kink the slope is always positive.
Therefore, the background soliton approximation can fail
drastically for large g/φ0.
Acknowledgements A.M. and A.A. thank E. R. Bezerra de Mello for
the helpful discussions. The authors thank Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) for the financial support.
Also, A.M. thanks PNPD/CAPES for the partial support.
Open Access This article is distributed under the terms of the Creative
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1. N.J. Zabusky , M.D. Kruskal , Phys. Rev. Lett . 15 , 240 ( 1965 )
2. D.J. Korteweg , F. de Vries, Philos. Mag. 39 , 422 ( 1895 )
3. R. Rajaraman , Solitons and Instantons ( North-Holland, Amsterdam, 1982 )
4. P. Drazin , R. Johnson, Solitons: An Introduction (Cambridge University Press, Cambridge, 1996 )
5. N. Manton , P. Sutcliffe , Topological Solitons (Cambridge University Press, Cambridge, 2004 )
6. H. Watanabe , H. Murayama, Phys. Rev. Lett . 112 , 191804 ( 2014 )
7. R. Driben , Y.V. Kartashov , B.A. Malomed , T. Meier , L. Torner, Phys. Rev. Lett . 112 , 020404 ( 2014 )
8. K.V. Samokhin , Phys. Rev. B 89 , 094503 ( 2014 )
9. E.R. Bezerra de Mello , A.A. Saharian , JHEP 04 , 046 ( 2009 )
10. A.I. Milstein , I.S. Terekhov , U.D. Jentschura , C.H. Keitel , Phys. Rev. A 72 , 052104 ( 2005 )
11. T.H.R. Skyrme , Proc. R. Soc. Lond. A 247 , 260 ( 1958 )
12. T.H.R. Skyrme , Proc. R. Soc. Lond. A 252 , 236 ( 1959 )
13. T.H.R. Skyrme , Proc. R. Soc. Lond. A 260 , 127 ( 1961 )
14. T.H.R. Skyrme , Nucl. Phys. 31 , 556 ( 1962 )
15. T.H.R. Skyrme , J. Math. Phys. 12 , 1735 ( 1971 )
16. W.A. Bardeen , M.S. Chanowitz , S.D. Drell , M. Weinstein , T.-M. Yan , Phys. Rev. D 11 , 1094 ( 1975 )
17. R. Friedberg , T.D. Lee , Phys. Rev. D 15 , 1694 ( 1977 )
18. R. Friedberg , T.D. Lee , Phys. Rev. D 16 , 1096 ( 1977 )
19. A. Mohammadi , E.R. Bezerra de Mello , A.A. Saharian , Class. Quant. Grav. 32 , 135002 ( 2015 )
20. S.S. Gousheh , A. Mohammadi , L. Shahkarami , Eur. Phys. J. C 74 , 3020 ( 2014 )
21. F. Charmchi , S.S. Gousheh , Nucl. Phys. B 883 , 256 ( 2014 )
22. E.R. Bezerra de Mello , A.A. Saharian , Eur. Phys. J. C 73 , 2532 ( 2013 )
23. S.S. Gousheh , A. Mohammadi , L. Shahkarami, Phys. Rev. D 87 , 045017 ( 2013 )
24. L. Shahkarami , A. Mohammadi , S.S. Gousheh , JHEP 11 , 140 ( 2011 )
25. R. Jackiw , C. Rebbi , Phys. Rev. D 13 , 3398 ( 1976 )
26. M. Shifman , Advanced Topics in Quantum Field Theory (Cambridge University Press, Cambridge, 2012 )
27. V.A. Rubakov , M.E. Shaposhnikov , Phys. Lett. B 125 , 136 ( 1983 )
28. M.V. Libanov , S.V. Troitsky , Nucl. Phys. B 599 , 319 ( 2001 )
29. J.M. Frere , M.V. Libanov , S.V. Troitsky , Phys. Lett. B 512 , 169 ( 2001 )
30. S. Randjbar-Daemi , M. Shaposhnikov , JHEP 04 , 016 ( 2003 )
31. W. Nahm , D.H. Tchrakian , JHEP 04 , 075 ( 2004 )
32. A. Melfo , N. Pantoja , J.D. Tempo , Phys. Rev. D 73 , 044033 ( 2006 )
33. L. Shahkarami , S.S. Gousheh , JHEP 06 , 116 ( 2011 )
34. V.A. Gani , V.G. Ksenzov , A.E. Kudryavtsev , Phys. Atom. Nucl . 73 , 1889 ( 2010 )
35. V.A. Gani , V.G. Ksenzov , A.E. Kudryavtsev , Phys. Atom. Nucl . 74 , 771 ( 2011 )
36. F. Charmchi , S.S. Gousheh , Phys. Rev. D 89 , 025002 ( 2014 )
37. R. Jackiw , S.-Y. Pi, Phys. Rev. Lett . 98 , 266402 ( 2007 )
38. N. Graham , R.L. Jaffe , Nucl. Phys. B 544 , 432 ( 1999 )
39. Shi-Hai Dong , Xi-Wen Hou , Zhong-Qi Ma , Phys. Rev. A 59 , 995 ( 1999 )
40. Shi-Hai Dong , Xi-Wen Hou , Zhong-Qi Ma , Phys. Rev. A 58 , 2160 ( 1998 )
41. M.A. Shifman , A.I. Vainshtein , M.B. Voloshin , Phys. Rev. D 59 , 045016 ( 1999 )
42. F. Charmchi , S.S. Gousheh , S.M. Hosseini , J. Phys . A Math. Theor . 47 , 335401 ( 2014 )